Mathematics Grade 8 Part 1

Teacher’s Note and Reminders 4. y 5. y (2, 4) (2, 2) Don’t 3 4 Forget! x 2 3Discuss the trend of the graph and differentiate the lines whose slopes are 23positive (m > 0) and negative (m < 0) as well as the lines whose slopes are zero 1 2and undefined. 0 x 1 -2 -1 01 2 0 -1 -2 -1 01 -2 (0, -2) -1 -2 -3 QU ?E S T I ONS 1. How did you find the slope of the line? 2. What is the trend of the graph? Is it increasing? Or decreasing? 3. What is the slope of each increasing graph? What are the signs of the slopes? 4. What is the slope of the decreasing graph? What is the sign of the slope? 5. Do the graphs represent linear functions? Why or why not? 6. What is the slope of the horizontal line? How about the vertical line? Note that:  A basic property of a line, other than vertical line, is that its slope is constant.  The slope of the horizontal line is zero while that of the vertical line is undefined. Both lines do not represent linear functions.  The value of the slope m tells the trend of the graph. • If m is positive, then the graph is increasing from left to right. • If m is negative, then the graph is decreasing from left to right. • If is zero, then the graph is a horizontal line. • If ym is undefined, then the graph is a vertical line. m yy yAnswers to Challenge Questions: xx x x1. a. a = - 7/2 b. a = - 1/5 Challenge Questions2. The slope is - A/B 1. Determine the value of a that will make the slope of the line through the two given points equal to the given value of m. a. (4, -3) and (2, a); m = 1 4 b. (a + 3, 5) and (1, a – 2); m = 4 2. If A, B, and C∈ℜ and the line is described by Ax + By = C, find its slope. 200

Teacher’s Note and Reminders y Consider the graph of the function f defined 5 (2, 5) f(x) = 2x + 1 at the right. 4 Question to Ponder: 3 1. What is the slope of the line using any of the formulae? 2 2. Compare the slope you have computed to 1 (0, 1) x the numerical coefficient of x in the given function 0 -3 -2 -1 0 1 2 3 4 -1 -2 -3 -4 The slope of the function f defined by f(x) = mx + b is the value of m. Exercise 7 Determine the slope of each line, if any. Identify which of the lines is vertical or horizontal. 1. f(x) = 2x – 5 6. 2x – y = 5 Don’t 2. f(x) = -3x + 7 7. 7x – 3y – 10 = 0 Forget! 3. f(x) = x + 6 8. 1 x + 1 y – 8 = 0 24 4. f(x) = 1 x – 8 9. x = 8 4 5. f(x) = 2 x – 1 10. 2y + 1 = 0 32Answers to Exercise 7: Activity 9 STEEP UP!1. m = 2 6. m = 2 Description: This activity will enable you to use the concept of slope in real life. This2. m = -3 7. m = 7/3 Direction: can be done by group of 5 members.3. m = 1 8. m = -2 Find any inclined object or location that you could see in your school and4. m = 1/4 9. m is undefined; vertical line then determine its steepness.5. m = 2/3 10. m = 0; horizontal line QU ?E S T I ONS 1. How did you find the steepness of the inclined object?Ask the students to perform Activity 9. Allow them to go out of the classroom and 2. Have you encountered any difficulty in determining the steepnesslook for any inclined object. Let them find its slope. This will enable them to seethe connection of the concept to real life. of the object? Explain your answer. 201

Discuss graphing linear equations. Start with any two points. Let them recall Graphs of Linear Equationsthat a linear equation is an equation whose graph is a line and in Geometry, twopoints determine a line. That’s why, two points are sufficient to draw the graph You have learned earlier that a linear function can be described by its equation,of a linear equation. either in the form y = mx + b or Ax + By = C. A linear equation can also be described by itsAnswers to Exercise 8: graph. Graphing linear equations can be done using any of the four methods:1. 3. 1. Using two points 2. Using x- and y-intercepts2. 4. 3. Using the slope and the y-intercept 4. Using the slope and a point Teacher’s Note and Reminders Using Two Points Don’t Forget! One method of graphing a linear equation is using two points. In Geometry, you learned that two points determine a line. Since the graph of the linear equation is a line, thus two points are enough to draw a graph of a linear equation. y Illustrative Example 5 Graph the function y = 2x + 1. 4 You may assign any two values for x, say 0 and 1. By substitution, 3 (1, 3) 2 1 (0, 1) y = 2x + 1 y = 2x + 1 -3 -2 -1 0 1 2 3 4x y = 2(0) + 1 y = 2(1) + 1 -1 y = 0 + 1 y = 2 + 1 y = 1 y = 3 -2 -3 -4 If x = 0, then y = 1. Furthermore, if x = 1, then y = 3. So, the ordered pairs are (0, 1) and (1, 3). This means that the line passes through these points. After finding the ordered pairs of the two points, plot and connect them. Your output is the graph of the linear equation. Exercise 8 Graph each linear equation that passes through the given pair of points. 1. (1, 2) and (3, 4) 3. (-2, 5 ) and ( 1 , - 1 ) 2. (5, 6) and (0, 11) 3 23 Using x-Intercept and y-Intercept 4. (- 1 , - 1 ) and ( 3 , 1 ) 35 22 Secondly, the linear equation can be graphed by using x-intercept a and y-intercept b. The x- and y-intercepts of the line could represent two points, which are (a, 0) and (0, b). Thus, the intercepts are enough to graph the linear equation. 202

Discuss graphing linear equations using x- and y-intercepts. Emphasize that To graph the equation y = 2x + 1 using this method, you need to solve the x-interceptx-intercept a is the abcissa of the coordinates of the point (a, 0) that intersects by letting y = 0 and the y-intercept by letting x = 0.the x-axis while y-intercept b is the ordinate of the coordinates of the point (0, b)that intersects the y-axis. This means that two points exist to represent x- and Letting y = 0, the equation y = 2x + 1 becomes yy-intercepts. Thus, x-intercept and y-intercept are sufficient to draw the graph ofthe linear equation. 0 = 2x + 1 Substitution 5Processes on how to solve for x- and y-intercepts are provided. Links are alsoprovided for further references. -2x = 1 Addition Property of Equality 4Answers to Exercise 9: x = - 1 1. 3. 2 Multiplication Property of Equality 3 y-intercept2. 4. Letting x = 0, y = 2x + 1 becomes 2 y = 2(0) + 1 Substitution y = 0 + 1 Simplification - 1 11 2 -3 -2 -1 0 1 4x 2 3 y = 1 Simplification -1 x-intercept -2 The x-intercept a is - 1 while the y-intercept b is 1. -3 2 -4 Now, plot the x- and y-intercepts, then connect them. The x-intercept is the abscissa of the coordinates of the point in Web Links which the graph intersects the x-axis. However, the y-intercept is the Click these links for further ordinate of the coordinates of the point in which the graph intersects the y-axis. references: 1. http://www.youtube.com/ watch?v=mvsUD3tDnHk &feature=related. 2. http://www.youtube.com/ watch?v=mxBoni8N70Y Exercise 9 Graph each linear equation whose x-intercept a and y-intercept b are given below. 1. a = 2 and b = 1 3. a = -2 and b = -7 2. a = 4 and b = -1 4. a = 1 and b = -2 Using Slope and y-Intercept 2 The third method is by using the slope and the y y-intercept. This can be done by identifying the slope and the y-intercept of the linear equation. 5 4 run = 1 3 (1, 3) In the same equation y = 2x + 1, the slope m is 2 rise = 2 2 1 2 3 4x 1 and y-intercept b is 1. Plot first the y-intercept, then use the slope to find the other point. Note that 2 means 2 , y-intercept which means rise = 2 and run = 1. Using the y-interce1pt -3 -2 -1 0 -1 -2 as the starting point, we move 2 units upward since -3 rise = 2, and 1 unit to the right since run = 1. -4 203

Discuss graphing linear equations using slope and y-intercept. A y-intercept Web Links Note that if rise is less than zero (or negative), we moverepresents a point. Thus, it is necessary to find for another point. That could be Click these links for moredone by using the slope and y-intercept. downward from the first point to look for the second point.Processes on how to find another point using slope and y-intercept are provided. examples:Links are also provided for further references. Similarly, if run is less than zero (or negative), we move to theAnswers to Exercise 10: 1. http://www.youtube.com/1. 3. watch?v=QIp3zMTTACE left from the first point to look for the second point. Moreover, a2. 4. 2. http://www.youtube.com/ negative rational number - 1 can be written as either -1 or 1 but watch?v=jd-ZRCsYaec 3. http://www.youtube.com/wa tch?v=EbuRufY41pc&featur e=related not -1. 2 2 -2 -2 Exercise 10 Graph each linear equation given slope m and y-intercept b. 1. m = 2 and b = 3 3. m = 1 and b = 3 y 2. m = 1 and b = 5 2 Using Slope and One Point 4. m = -3 and b = - 3 2 The fourth method in graphing linear equation is by using 4 the slope and one point. This can be done by plotting first the 3 given point, then finding the other point using the slope. 2 B (0, 1) 1 02 3x The linear equation y = 2x + 1 has a slope of 2 and a -3 -2 -1 01 2 point (-1, -1). To find a point from this equation, we may assign A (-1, -1) -1 any value for x in the given equation. Let’s say, x = -1. The 1-2 value of y could be computed in the following manner: -3 -4 y = 2x + 1 Given y = 2(-1) + 1 Substitution y = -2 + 1 Simplification y = -1 Simplification Complete the statement below: The line passes through the point _____. The point found above is named A whose coordinates are (-1, -1). Since the slope of the line is 2 which is equal to 2 , use the rise of 2 and run of 1 to determine the coordinates 1 of B (refer to the graph). This can also be done this way. Web Links B = (-1 + 1, -1 + 2) = (0, 1) Use this link http:// Note that 2 (the rise) must be added to the y-coordinate while 1 (the run) must be added to x-coordinate. www.youtube.com/ watch?v=f58Jkjypr_I which is a video lesson for another example. 204

Discuss graphing linear equations using slope and one point. It is necessary to Exercise 11find for another point. That could be done by using the slope and one point. Graph the following equations given slope m and a point.Processes on how to find another point using slope and a point are provided.Links are also provided for further references. 1. m = 3 and (0, -6) 3 m = 1 and (0, 4) 2. m = -2 and (2, 4) 2Answers to Exercise 11: Activity 10 WRITE THE STEPS 4. m = 3 and (2, -3)1. 3. 2 Description: This activity will enable you to summarize the methods of graphing a Direction: linear equation. Fill in the diagram below by writing the steps in graphing a linear equation using 4 different methods.2. 4. Using Two Points Using x- and y-Intercepts Using Slope and y-Intercept Using Slope and One Point QU ?E S T I ONS 1. Among the four methods of graphing a linear equation, which one is easiest for you? Justify your answer. 2. Have you encountered any difficulty in doing any of the four methods? Explain your answer. 205

Assess students’ knowledge about the steps on drawing the graph of the linear Activity 11 MY STORY yequation using the four methods. Allow them to go back to how these methodsare done. 50Allow the students to create their own story about the given graph in performing Description: This activity will enable you to 40 (4, 40)Activity 11. Varied answers to this activity are expected. Direction: analyze the graph and connect this to real life. 30 (3, 30)Let the students describe the graph of the linear function using its x-intercept, Create a story out of the graph of they-intercept, slope, trend and equation. You may give additional graph for further linear equation at the right. Share 20 (2, 20)practice. Ask them to answer Activity 12. this to your classmate. 10 (1, 10) 0 (0, 0) x -1 0 1 2 3 4 5 -10Answers to Activity 12: QU QU?E S T I ONS NS 1. Do you have the same story with your classmates? 2. Is your story realistic? Why?1. x-intercept: -32. y-intercept: 2 Activity 12 DESCRIBE ME (PART III)!3. rise: 34. run: 2 Description: This activity will enable you to describe the 25. slope: 3/2 graph of a linear equation in terms of its6. trend: increasing -3 Table: 0 intercepts, slope and points. 1 x0 Direction: Given the graph at the right, find the 0 y2 following: -2 -1 0 -1 -2 -3 1. x-intercept 4. run -1Teacher’s Note and Reminders 2. y-intercept 5. slope -2 3. rise 6. trend -3 Complete the table below: -4 x y Don’t ?E S T I O 1. How did you identify the x-intercept and y-intercept? Forget! 2. In your own words, define x-intercept and y-intercept. 3. How did you find the rise and run? 4. How did you find the slope? 5. Is it increasing or decreasing from left to right? Justify your answer. 6. Have you observed a pattern? 7. What happen to the value of y as the value of x increases? 8. How can the value of a quantity given the rate of change be predicted? 206

Discuss finding the equation of the line. Start it by using the slope –intercept form Finding the Equation of the Liney = mx + b. The equation of the line can be determined using the following formulae:Answers to Questions of Activity 13: a. slope-intercept form: y = mx + b; b. point-slope form: y – y1 = m(x – x1); and1. The value of m in each equation c. two-point form: y – y1 = y2 – y1 (x – x1).2. The value of b in each equation x2 – x13. Activity 13 SLOPE AND Y-INTERCEPTEquation of the Line Slope y-Intercept Description: This activity will enable you to find the equation of the line using slope-a. y = 2x 2 0 intercept form. 4 -5 Materials: graphing paper 5b. y = 2x + 4 2 4 pencil or ballpenc. y = 2x – 5 2 Direction: Graph these equations in one Cartesian plane.d. y = x + 5 1 a. y = 2x c. y = 2x – 5 e. y = -2x + 4e. y = -2x + 4 -2 b. y = 2x + 4 d. y = x + 54. The value of m is the slope of the line y = mx + b and the value of b is its QU ?E S T I ONS 1. What is the slope of each line? Use the formula m = rise to answer y-intercept. run this question.5. The slope of y = 7x + 1 is 7 while its y-intercept is 1. 2. What is the y-intercept of each line? Teacher’s Note and Reminders 3. Complete the table below using your answers in 1 and 2. Equation of the Line Slope y-Intercept a. y = 2x b. y = 2x + 4 c. y = 2x – 5 d. y = x + 5 e. y = -2x + 4 Don’t 4. What can you say about the values of m and b in the equation y = Forget! mx + b and the slope and the y-intercept of each line? Write a short description below. ____________________________________________________ 5. Consider the equation y = 7x + 1. Without plotting points and computing for m, what would you expect the slope to be? How about the y-intercept? Check your answer by graphing. Are your expectations about the slope and the y-intercept of the line correct? Example: Find the equation of the line whose slope is 3 and y-intercept is -5. Solution: The equation of the line is y = 3x – 5. 207

Assess the students’ knowledge about the slope and the y-intercept of the line Slope-Intercept Form of the Equation of a Linewhose equation is in the form Ax + By = C. Do not give this as an assignment. The linear equation y = mx + b is in slope-intercept form. The slope of the line is mAnswers to Activity 14: and the y-intercept is b.1. 2x + 5 y = 10 m = − 2 b = 2 Activity 14 FILL IN THE BOX 5 Description: This activity will assess what you have learned in identifying the slope Direction: and y-intercept of the line whose equation is in the form Ax + By = C.2. 3 x – 6y = 7 Complete the boxes below in such a way that m and b are slope and m = 1 b = − 7 y-intercept of the equation, respectively. You are allowed to write the numbers 1 to 10 once only. 26 1. 2x + y = 3. 3 x + y=13. 3 6 x + 9 y = 1 8 m = − b = 2 m = − b=2 m = − 4 b = 2 5Answers to Activity 15: 1. m = yx22 – y1 2. x – 6y = 7 – x1 m = b = −2. Since the two points of the line are represented by (x1 – y1) and (x, y), its 26 y – y1 slope is m = x – x1 Activity 15 THINK-PAIR-SHARE3. Two-Point Form of the linear equation Description: This activity will enable you to generate 5 Direction: y – y1 Point-Slope Form of the equation of a 44. m = line. Shown at the right is a line that (x, y) contains the points (x1, y1) and (x, y). x – x1 Note that the (x1, y1) is a fixed point 3 y – y1 = m(x – x1 ) on the line while (x, y) is any point contained on the line. 2 Give what are asked. 1 (x1, y1) -3 -2 -1 0 1 2 3 4 -1 -2 1. Recall the formula for slope -3 given two points. -4 2. How do you compute the slope of this line? 3. What formula did you use? 4. Solve for the Point-Slope Form of a line by completing the following: m= y– Why? x – y – = m(x – ) 208

Let the students derive the formula of Point-Slope and Two-Point Forms of the Point-Slope Form of the Equation of a Linelinear equation. Let each student find a partner and discuss Activities 15 and16. You may also assist each pair of students in the derivation of the formulae. The linear equation y – y1 = m(x – x1) is the point-slope form. The value of m is theDiscuss their answers and give examples. slope of the line which contains a fixed point P1(x1, y1).Answers to Exercise 12: Exercise 12 Find the equation of the line of the form y = mx + b given the slope and a point.1. y = 2x + 4 6. y = 1/2x – 3 1. m = 2; (0, 4) 6. m = 1 ; (-6, 0)2. y = x – 7 7. y = 2/3x + 8 2. m = 1; (5, -2) 23. y = -5x – 6 8. y = - 7/2x – 11 3. m = -5; (-3, 9) 4. y = -7x + 27 9. y = - 7/4x + 9/2 4. m = -7; (4, -1) 7. m = 2 ; (0, 8)5. y = -x + 9 10. y = 1/2x + 35/12 5. m = -1; (7, 2) 3Answers to Activity 16: Activity 16 THINK-PAIR-SHARE 8. m = - 7 ;(-4, 3) 2 y2 – y1 x2 – x1 9. m = - 7 ;(-2, 8) 4 10. m = 1 , (- 1 , 8 ) 2 231. m =2. y – y1 = m(x – x1) Description: This activity will enable you to derive the Two-Point Form of the equation Direction: of the line. Again, recall the formula for the slope and the Point-Slope3. Subsitute the formula of m to the Point-Slope Form. Form of the equation of the line. Answer the following guide questionsAnswers to Exercise 13: 1. Write the formula of slope m of the line given two points in the box.1. y = 3x – 5 6. y = - 1/4x + 1/2 2. Write the Point-Slope Form of the equation of the line in the box.2. y = -3x + 28 7. y = - 1/4x + 15/83. y= -x + 2 8. y = -4x – 9/24. y = -6x – 43 9. y = 1/35. y = 5x + 15 10. y = -7/12x + 1/24Teacher’s Note and Reminders 3. State the justification in the second statement below. y – y1 = m(x – x1) Point-Slope Form Don’t y – y1 = y2 – y1 (x – x1) Why? Forget! x2 – x1 Two-Point Form of the Equation of a Line The linear equation y – y1 = y2 – y1 (x – x1) is the Two-Point Form, where (x1, y1) and (x2, y2) are the coordinates of x2 – x1 P2, respectively. P1 and 209

Teacher’s Note and Reminders Exercise 13 Find the equation of the line of the form y = mx + b that passes through the following pairs of points. 1. (3, 4) and (4, 7) 6. (0, 1 ) and (1, - 1 ) 2. (8, 4) and (6, 10) 22 3. (3, -1) and (7, -5) 4. (-8, 5) and (-9, 11) 7. ( 7 , 1) and (- 1 , 2) 5. (-1, 10) and (0, 15) 22 8. (- 1 , - 5 ) and (- 3 , 3 ) 22 22 9. (-15, 1 ) and (- 1 , 1 ) 23 23 10. (- 5 , 3 ) and ( 1 , - 1 ) 22 24 To enrich your skills in finding the equation of the line, which is horizontal, vertical or slanting, go to this link http://www.mathplayground.com/SaveTheZogs/ SaveTheZogs_IWB.html. You can also visit the link in finding the equation of the line, where two points can be moved from one place to another http://www.mathwarehouse.com/algebra/ linear_equation/linear-equation-interactive-activity.php Activity 17 IRF WORKSHEET REVISITED Don’t Description: Below is the IRF Worksheet in which you will give your present knowledge Forget! Direction: about the concept. Give your revised answers of the questions provided in the first column andElicit present knowledge about the linear functions by answering the “Revised write them in the third column. Compare your revised answers from yourAnswer” column in the IRF Worksheet. Compare their revised answers of the initial answers.questions to their initial answers. Questions Initial Revised Final Answer Answer Answer 1. What is linear function? 2. How do you describe a linear function? 3. How do you graph a linear function? 4. How do you find an equation of the line? 5. How can the value of a quantity given the rate of change be predicted? 210

Provide students the opportunities to think deeper and apply their knowledge In this section, the discussions were about linear functions. Go back to theand skills in solving word problems involving linear functions. Flow chart and previous section and compare your initial ideas with the discussions. How much of yourguide questions are provided. Even though your little assistance is encouraged, initial ideas are found in the discussions? Which ideas are different and need revision?allow the students to solve the problems on their own. Ask the students to do Now you know the important ideas about this topic, let’s go deeper by moving on to theActivities 18, 19, and 20. next section.Answers to Activity 18: Distance 0 200 400 600 800 1000 WWhhaatt ttoo UUnnddeerrssttaanndd(in meters) 40 43.50 47 50.50 54 57.50 Your goal in this section is to take a closer look ate the real-life problems x involving linear equations and relations. Amount Activity 18 RIDING IN A TAXI (in Php) y1. The dependent variable is the amount because it depends on the distance. Description: This activity will enable you to solve real-life problems involving linear2. The independent variable is the distance because it controls the amount. functions.3. It represents a line.4. The y-intercept of the line is 40. Direction: Consider the situation below and answer the questions that follow.5. The slope is 7/400.6. The linear function f is f(x) = 7/400x + 40. Emman often rides a taxi from (a) Emman will have to pay Php 50.50. one place to another. The standard fare (b) He will have to pay Php 145. in riding a taxi is Php 40 as a flag down rate plus Php 3.50 for every 200 meters (c) If he pays Php 68, then he traveled 1600 meters or less than 1600 meters or a fraction of it. but greater than 1400 meters. If he pays Php 75, then he traveled 2000 meters or less than 2000 meters but greater than 1800 meters. If he Complete the table below: pays Php 89, then he traveled 2800 meters or less than 2800 meters but greater than 2600 meters. Finally, if he pays Php 92.50, then he traveled Distance 0 200 400 600 800 1000 3000 meters or less than 3000 meters but greater than 2800 meters. (in meters)7. The linear equation is 7x – 400y = -1600, instead of 7x – 800y = -1600. x Amount (in Php) y QU ?E S T I ONS 1. What is the dependent variable? Explain your answer. 2. What is the independent variable? Explain your answer. 3. Based on the completed table, would the relation represent a line? 4. What is the y-intercept? Explain your answer. 5. What is the slope? Explain your answer. 6. Write the linear function and answer the following questions. (a) If Emman rides a taxi from his workplace to the post office with an approximate distance of 600 meters, how much will he pay? 211

Answers to Activity 19: (a) If he rides a taxi from his residence to an airport with an approximate distance of 6 kilometers, how much will he pay?Step 1: The dog’s weight is 1 kg at birth. Its weight is 6 kg after a month. (b) If Emman pays Php 68, how many kilometers did he travel?Step 2: The dependent variable is the dog’s weight while the independent How about Php 75? Php 89? Php 92.50? variable is the time. 7. Write the equation of the line in the form Ax + By = C using yourStep 3: x0 123 answer in number 6.Step 4: 6 11 16 y1 8. Draw the graph of the equation you have formulated in item 7. y Activity 19 GERMAN SHEPHERD Description: This activity will enable you to solve problems involving linear functions by following the steps provided. Direction: Do the activity as directed. x You own a newly-born German shepherd. SupposeStep 5: The slope m = 5, y-intercept b = 1. The equation is y = 5x + 1. the dog weighs 1 kg at birth. You’ve known from your friend that the monthly average weight gained by the dog is 5 kg. If Teacher’s Note and Reminders the rate of increase of dog’s weight every month is constant, determine an equation that will describe the dog’s weight. Predict the dog’s weight after five months using mathematical equation and graphical representation. Complete the flow chart below then use it to answer the questions that follow. Don’t Forget! 212

Ask the students to answer Activity 20. Allow the them to use the flow chart QU ?E S T I ONS 1. What equation describes the dog’s weight?given in Activity 19. In answering item 1 of this activity, emphasize that x must 2. What method did you use in graphing the linear equation?be the time that exceeds after 3 minutes, or consider the domain is {x|x ≥ 3}. 3. How will you predict the dog’s weight given the rate of changeThis is important because if we fail to do it, the graph of the function is not a lineanymore. Give more real-life problems involving linear functions. in his weight?Answers to Activity 20: Activity 20 WORD PROBLEMS1. A caller will have to pay Php 10. Let x be the time that exceeds after 3 minutes Description: This activity will enable you to solve more word problems involving linear and let y be the charge. The rule is y = x + 5. Direction: functions. In this activity, you are allowed to use the flow chart given in Activity 19.2. The formula to be used in solving this problem is t = d/r or t = 1/r (d), where t Solve the following. Show your solutions and graphs. is time, r is rate and d is distance. Given in this problem are r = 60 kph, which is constant, and d = 240 kilometers. So, the rule in this problem is t = 1/60(d). 1. A pay phone service charges Php 5 for the first three minutes and If d = 240 kilometers, then t = 4 hours. Php 1 for every minute additional or a fraction thereof. How much will a caller have to pay if his call lasts for 8 minutes? Write a rule that3. Let x be the number of donuts sold and let y be the total price. The rule that best describes the problem and draw its graph using any method. best describes the function is y = 18x + 5. It is assumed that there are 1 to 24 donuts sold; thus, the domain of the relation is the {x|1 ≤ x ≤ 24}. There would 2. A motorist drives at a constant rate of 60 kph. If his destination is be 12 donuts in the box whose price is Php 221. 240 kilometers away from his starting point, how many hours will it take him to reach the destination? Write a rule that best describes Teacher’s Note and Reminders the problem and draw its graph using any method. Don’t 3. Jolli Donuts charges Php 18 each for a special doughnut plus aForget! fixed charge of Php 5 for the box which can hold as many as 24 donuts. How many doughnuts would be in a box priced at Php 221? Write a rule that best describes the problem and draw its graph. In your graph, assume that only 1 to 24 doughnuts are sold. Activity 21 FORMULATE YOUR OWN WORD PROBLEM! Description: This activity will enable you to formulate your own word problem involving linear functions and to answer it with or without using the 5-step procedure. Direction: Formulate a word problem involving linear functions then solve. You may or may not use the flow chart to solve the problem. Be guided by the given rubric found in the next page. 213

Let the students formulate real-life problems involving linear function in Activity QU ?E S T I ONS 1. What equation describes the dog’s weight?21. This activity can be done by groups of five members each. Allow students touse the 5-step procedure of the flow chart provided in the previous activity. D2i.d youWehantcmoeuthnotderdiadnyyouduifsfeicinulgtyrapinhinfogrtmheulilnaetainr geqrueaatiol-nli?fe problem 3. Hionwvowlvililnygoulinpereadricfut nthcetiodongs’?s wEexipghlatignivyeonutrhaenrastwe eorf. change in his Teacher’s Note and Reminders weight? Don’t Forget! RUBRIC: PROBLEMS FORMULATED AND SOLVED Score Descriptors Poses a more complex problem with 2 or more correct possible solutions and 6 communicates ideas clearly; shows in-depth comprehension of the pertinent concepts and/or processes and provides explanations wherever appropriate. Poses a more complex problem and finishes all significant parts of the solution and 5 communicates ideas clearly; shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes all significant parts of the solution and 4 communicates ideas clearly; shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes most significant parts of the solution and 3 communicates ideas clearly; shows comprehension of major concepts although neglects or misinterprets less significant ideas or details. 2 Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension. 1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach. Activity 22 YOU ARE THE SCHOOL PRINCIPAL This is a preparatory activity which will lead you to perform well the transfer task in the next activity. This can be a group work.This activity is a Scaffold Level 3 of the transfer task. A little of your guidance is Situation:important in order for students’ to be ready to perform the final task in Activity 24. You are the school principal of a certain school. Every week you conduct an information drive on the different issues or concerns in your school through announcements during flag ceremony or flag retreat or during meetings with the department heads and teachers. For this week, you noticed that water consumption is high. You will make and present an informative leaflet with design to the members of the academic community. In your leaflet design, you must clearly show water bill and water consumption and how these two quantities relate each other. The leaflet must also reflect data on the quantity of water bill for the previous five months, and a detailed mathematical computation and a graphical presentation that will aid in predicting the amount of water bill that the school will pay. 214

Elicit students’ present knowledge of Linear Functions by answering the “Final recommendations to save water. Your leaflet as a whole will be assessed usingAnswer’ column in the IRF Worksheet. Compare their final answers to their initial the following criteria: use of appropriate mathematical concepts and accuracy,and revised answers. organization, quality of presentation, and practicality of recommendations.Before giving the transfer task, ask first the students if they have realizationsabout the topic. Also, ask them the question: What new connections have you Activity 23 IRF WORKSHEET REVISITEDmade for yourself? Then say, now that you have a deeper understanding of thetopic, you are ready to do the task in the next section. Description: Below is the IRF Worksheet in which you will write your present knowledge about the concept. Teacher’s Note and Reminders Direction: Complete IRF sheet below. Questions Initial Revised Final Answer Answer Answer 1. What is linear function? 2. How do you describe a linear function? 3. How do you graph a linear function? 4. How do you find an equation of the line? 5. How can the value of a quantity given the rate of change be predicted? Don’t What new realizations do you have about the topic? What new connections haveForget! you made for yourself? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section.WWhhaatt ttooTTrraannssffeerr WWhhaatt ttooTTrraannssffeerr Allow the students to perform Activity 24 without your assistance, if possible. Your goal in this section is to apply your learning to real-life situations. You will Rate their performance based on the criteria of the rubric provided. be given a practical task which will demonstrate your understanding. 215

Teacher’s Note and Reminders Activity 24 YOU ARE A BARANGAY COUNCILOR Don’t This activity is the transfer task. You have to perform this in your own Forget! community. Situation: You are a barangay councilor in San Sebastian. Every month, you conduct in- formation drive on the different issues that concern every member in the community. For the next month, your focus is on electricity consumption of every household. You are tasked to prepare a leaflet design which will clearly explain about electricity bill and consumption. You are to include recommendations to save water. You are expected to orally present your design to the other officials in your barangay. Your output will be assessed according to the rubric below. RUBRIC: LEAFLET DESIGN CRITERIA Exemplary Satisfactory Developing Beginning 4 3 2 1 The mathematical The mathematical The mathematical The mathematical Use of concepts used concepts used concepts used concepts used mathematical concepts and are correct and are correct and are correct but are wrong and the accuracy the computations the computations the computations computations are are accurate. are accurate. are inaccurate. inaccurate. Brief explanation is provided. Organization The ideas The ideas The ideas The ideas and and facts are and facts are and facts are facts are not well complete, orderly completely mostly orderly presented. presented, and orderly presented. well prepared. presented. The presentation The presentation The presentation The presentation uses appropriate uses appropriate Quality of and creative visual designs. uses some visual does not include presentation visual designs. designs which are any visual inappropriate. design/s. The The Some The recommendations recommendations recommendations recommendations Practicality of are sensible, are sensible and are sensible and are insensible and recommendations doable and new doable. doable. undoable. to the community. You have just completed this lesson. Before you go to the next lesson, you have to answer the post-assessment. 216

POST-ASSESSMENT1. What is abscissa? a. It is a y-coordinate. b. It is a x-coordinate. c. It is a point on the xy-plane. d. It divides the plane into four regions called quadrant.2. Which best describes the point (3, -4)? a. It is 4 units above the x-axis and 3 units to the left of the y-axis. b. It is 4 units below the x-axis and 3 units to the left of the y-axis. c. It is 4 units above the x-axis and 3 units to the right of the y-axis. d. It is 4 units below the x-axis and 3 units to the right of the y-axis.3. Which relation below does NOT define a function? a. X Y c. X Y 5 a1 -5 8b 2 5 3 .cd4 8 b. X 1Y.1 d. X Y 5 1.5 1 1 1.4 2 1.6 3 4 217

4. What is the range of the relation at the right? a. {x|-3 ≤ x ≤ 3, x ∈ ℜ} b. {x|-3 < x < 3, x ∈ ℜ} c. {x|-3 ≤ x ≤ 3, x ∈ Z} d. {x|-3 < x < 3, x ∈ Z}5. The correct table of the function f defined by f(x) = 3x + 1 is xy a. x y b. x y c. x y d. -2 -5 -1 -3-2 -5 -2 -5 -2 -6 00 1301 -1 1 0 -3 2627 07 204 13 1 13 436 19 2 19 666. What is the equation of the line at the right? a. x + y = 1 c. 2x + y = 1 b. x – y = 1 d. 2x – y = 17. Find the equation of the line passing through the point (-3, 5) and whose slope is 2? a. y = 2x – 1 c. y = 2x + 8 b. y = 2x + 2 d. y = 2x + 11 218

8. Three steps to rewrite 3x – 4y = 7 into y = mx + b are shown below. What is the correct order of these steps? a. II, III, I c. III, II, I b. I, II, III d. II, I, III9. Which line in the figure at the right has a slope of zero? a. line l b. line m c. line n d. line p10. What will happen to the value of y in the equation 2x + 3y = 12 when the value of x decreases? a. The value of y will increase. b. The value of y will decrease. c. The value of y will not change. d. The value of y cannot be determined.11. John rode a taxi from a bus terminal to JB Mall whose distance is approximately four kilometers. After riding, he paid an amount of ₱110. Which variable is dependent? a. taxi riding b. the amount paid c. the distance travelled d. the person riding the taxi 219

For item numbers 12 and 13, refer to the situation below. The height h of the candle in centimeters is a function of time t in hours it has been burning. It is described by the table below: t012345 h(t) 10 8 6 4 2 012. Write the linear function h described by the table above? a. h(t) = 2t – 10 c. h(t) = 10 – 2t v. h(t) = 2t + 10 d. h(t) = 10 – t13. How long will it take the candle be completely melted? a. 3 b. 4 c. 5 d. 6 14. Find the slope of the roof indicated at the right. 4 ft a. 4/5 b. 5/4 10 ft c. 2/5 d. 5/2 For items 15 to 18, refer to the situation below: Jose, who is the SSG Business Manager, was given the task by the SSG President to canvass for a tarpaulin printing. He knew that in printing ad, the charge of tarpaulin printing is Php 12 per square foot and Php 100 for the layouting. 220

15. Which of the following equations best represents the total cost y with x number of square feet including layouting fee? a. y = 12x – 100 c. y = 100x – 12 b. y = 12x + 100 d. y = 100x + 12 16. What qualities you must look into in tarpaulin printing? I. The printing and layouting cost II. The quality of the printing output III. The brand of the PC used in layouting IV. The quality of the layout artist’s output a. I and II only c. I, II and IV only b. I, II and III only d. I, II, III and IV 1 7. Wa. hich o fy the follow ing b est re prese nts thce. relationship of the total cost y and the x number of square feet? y x x b. y d. y x 221

18. The SSG President told Jose that the dimensions of the tarpaulin are 5 feet by 4 feet. How many square feet is the tarpaulin? How much should Jose pay for the printing ad? a. 20 square feet; Php 420 c. 9 square feet; Php 320 b. 20 square feet; Php 340 d. 9 square feet; Php 208 For items 19 to 20, refer to the situation below. In a certain barangay, you are elected as the “Punong Barangay.” Hon. Bacus, who is a councilor, was assigned as the chairman of Committee on Energy. You gave him a task to make a Powerpoint presentation illustrating the relationship between electric bill and power consumption and to provide recommendations and friendly reminders to help minimize energy consumption.19. As a “Punong Barangay,” what criteria should you consider to assess Hon. Bacus’ PowerPoint presentation to ensure good quality of the delivery of presentation? I. colors and attractiveness II. content and delivery III. layout and design IV. font and font size used in the texts a. I only b. II and III only c. III and IV only d. II, III and IV20. If Hon. Lapuz has to choose one best representation of the relationship between electric bill and power consumption in his powerpoint presentation, what do you expect John should use to present his ideas in the clearest way? a. graph b. table c. mapping diagram d. rule or equationAnswer Key: 11. B 16. C1. B 6. A 12. C 17. C2. B 7. D 13. C 18. B3. B 8. C 14. C 19. D4. C 9. B 15. B 20. A5. A 10. A 222

SUMMARY/SYNTHESIS/GENERALIZATION This module was about relations and functions. It involved three lessons, namely: Rectangular Coordinate System,Representations of Relations and Functions, and Linear Functions. In the first lesson, students were expected to properly plot points in Cartesian plane and apply this to real life. In thesecond lesson, students were exposed to the different types of representing relations and functions. They were able to differentiatea function from a relation. Finally, in the last lesson, students were expected to solve, graph, and write in different ways a linear function. Moreimportantly, they were given the chance to formulate real-life problems, solve these using a variety of strategies, and demonstratetheir understanding of the lesson by doing some practical tasks.GLOSSARYCartesian plane Also known as the Rectangular Coordinate System which is composed of two perpendicular number lines (verticaland horizontal) that meet at the point of origin (0, 0).degree of a function f The highest exponent of x that occurs in the function f.dependent variable The variable (usually) y that depends on the value of the independent variable (usually) x.domain of the relation The set of first coordinates of the ordered pairs.function A relation in which each element in the domain is mapped to exactly one element in the range.function notation A notation in which a function is written in the form f(x) in terms of x.horizontal line A line parallel to the x-axis.independent variable The variable (usually) x that controls the value of the dependent variable (usually) y.line A straight line in Euclidean Geometry.Linear Function A function of first degree in the form f(x) = mx + b, where m is the slope and b is the y-intercept. 223

mapping diagram A representation of a relation in which every element in the domain corresponds to one or more elements in therange.mathematical phrase An algebraic expression that combines numbers and/or variables using mathematical operators.ordered pair A representation of point in the form (x, y).point-slope form The linear equation y − y1 = m(x − x1) is the point-slope form, where m is the slope and x1 and y1 are coordinatesof the fixed point.quadrants The four regions of the xy-plane separated by the x- and y-axes.range of the relation The set of second coordinates of the ordered pairs.rate of change The slope m of the line and is the quotient of change in y-coordinate and the change in x-coordinate.Rectangular Coordinate System Also known as Cartesian plane or xy-planerelation Any set of ordered pairs.slope of a line Refers to the steepness of a line which can be solved using the formulae: rise y2 − y1 . x2 − x1m = run or m =slope-intercept form The linear equation y = mx + b is in slope-intercept form, where m is the slope and b is the y-intercept.standard form The linear equation in the form Ax + By = C, where A, B and C are real numbers.trend Tells whether the line is increasing or decreasing and can be determined using the value of m (or slope).two-point form The linear equation y − y1 = y2 − y1 (x − x1) is the two-point form, where x1 and y1 are coordinates of the first point x2 − x1while x2 and y2 are coordinates of the second point. 224

vertical line A line parallel to the y-axis.Vertical Line Test If every vertical line intersects the graph no more than once, the graph represents a function.x-axis The horizontal axis of the Cartesian plane.x-intercept The x-coordinate of the point at which the graph intersects the x-axis.y-axis The vertical axis of the Cartesian plane.y-intercept The y-coordinate of the point at which the graph intersects the y-axis.REFERENCESDolciani, M. P., Graham, J. A., Swanson, R. A., Sharron, S. (1986). Algebra 2 and Trigonometry. Houghton Mifflin Company, OneBeacon Street, Boston, Massachussetts.Oronce, O. A., Mendoza, M. O. (2003). Worktext in Mathematics for Secondary Schools: Exploring Mathematics (ElementaryAlgebra). Rex Book Store, Inc. Manila, Philippines. Oronce, O. A., Mendoza, M. O. (2003). Worktext in Mathematics for Secondary Schools: Exploring Mathematics (IntermediateAlgebra). Rex Book Store, Inc. Manila, Philippines.Oronce, O. A., Mendoza, M. O. (2010). Worktext in Mathematics: e-math for Advanced Algebra and Trigonometry. Rex Book Store,Inc. Manila, Philippines.Ryan, M., et al (1993). Advanced Mathematics: A Precalculus Approach. Prentice-Hall, Inc., Englewood Cliffs, New Jersey.You Min, G.N. (2008). GCE “O” Level Pure Physics Study Guide. Fairfield Book Publishers: Singapore.http://hotmath.com/help/gt/genericalg1/section_9_4.htmlhttp://jongeslaprodukties.nl/yj-emilb.html 225

http://math.about.com/od/geometry/ss/cartesian.htmhttp://mathsfirst.massey.ac.nz/Algebra/StraightLinesin2D/Slope.htmhttp://members.virtualtourist.com/m/p/m/21c85f/http://people.richland.edu/james/lecture/m116/functions/translations.htmlhttp://roof-materials.org/wp-content/uploads/2011/09/Roof-Trusses.jpghttp://store.payloadz.com/details/800711-Other-Files-Documents-and-Forms-sports-car-.htmlhttp://wonderfulworldreview.blogspot.com/2011/05/mayon-volcano-albay-philippines.htmlhttp://www.dog-guides.us/german-shepherds/http://www.go2album.com/showAlbum/323639/coordinartiguana_macawhttp://www.mathtutor.ac.uk/functions/linearfunctionshttp://www.myalgebrabook.com/Chapters/Quadratic_Functions/the_square_function.phphttp://www.nointrigue.com/docs/notes/maths/maths_relfn.pdfhttp://www.onlinemathlearning.com/rectangular-coordinate-system.htmlhttp://www.plottingcoordinates.com/coordinart_patriotic.htmlhttp://www.purplemath.com/modules/fcns.htmhttp://www.teachbuzz.com/lessons/graphing-functionshttp://www.webgraphing.com/http://www.youtube.com/watch?NR=1&v=uJyx8eAHazo&feature=endscreenhttp://www.youtube.com/watch?v=EbuRufY41pc&feature=relatedhttp://www.youtube.com/watch?v=f58Jkjypr_Ihttp://www.youtube.com/watch?v=hdwH24ToqZIhttp://www.youtube.com/watch?v=I0f9O7Y2xI4http://www.youtube.com/watch?v=jd-ZRCsYaechttp://www.youtube.com/watch?v=mvsUD3tDnHk&feature=related.http://www.youtube.com/watch?v=mxBoni8N70Yhttp://www.youtube.com/watch?v=QIp3zMTTACEhttp://www.youtube.com/watch?v=-xvD-n4FOJQ&feature=endscreen&NR=1http://www.youtube.com/watch?v=UgtMbCI4G_I&feature=relatedhttp://www.youtube.com/watch?v=7Hg9JJceywA 226

TEACHING GUIDEModule 4: Linear Inequalities in Two VariablesA. Learning Outcomes Content Standard: The learner demonstrates understanding of key concepts of linear inequalities in two variables. Performance Standard: The learner is able to formulate real-life problems involving linear inequalities in two variables and solve these with utmost accuracy using a variety of strategies. UNPACKING THE STANDARDS FOR UNDERSTANDINGSUBJECT: LEARNING COMPETENCIESGrade 8 Mathematics 1. Differentiate between mathematical expressions andQUARTER mathematical equations.Second Quarter 2. Differentiate between mathematical equations and inequalities.STRAND: 3. Illustrate linear inequalities in two variables.Algebra 4. Graph linear inequalities in two variables on the coordinateTOPIC: plane.Linear Inequalities in Two Variables 5. Solve real-life problems involving linear inequalities in twoLESSONS: variables.1. Mathematical Expressions and Equations in Two ESSENTIAL UNDERSTANDING: ESSENTIAL QUESTION: Variables Students will understand that How can problems involving2. Equations and Inequalities in Two Variables3. Graphs of Linear Inequalities in Two Variables real-life problems where certain two quantities bounded by quantities are related and bounded conditions, restraints and by restraints, conditions and constraints be solved? constraints can be solved using linear inequalities in two variables. TRANSFER GOAL: Students will be able to apply the key concepts of linear inequalities in two variables in formulating and solving real-life problems. 227

B. Planning for AssessmentProduct/PerformanceThe following are products and performances that students are expected to come up with in this module.a. Linear inequalities drawn from real-life situations and the graph of eachb. Role-playing of real-life situations where linear inequalities in two variables are appliedc. Real-life problems involving linear inequalities in two variables formulated and solvedd. Budget proposal that demonstrates students’ understanding of linear inequalities in two variables. Assessment Map KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCE TYPE Pre-Test: Part I Pre-Test: Part I Pre-Test: Part I Graphing linear Solving problems Pre-Test: Part I Pre – assessment/ Identifying and inequalities in two involving linear Products and Diagnostic describing linear variables inequalities in two performances related inequalities in two variables to or involving linear variables and their Finding the solution set inequalities in two graphs of linear inequalities in Representing situations variables two variables using linear inequalities in two variables 228

Quiz: Lesson 1 Quiz: Lesson 1 Quiz: Lesson 1 Identifying linear Graphing linear Representing situations inequalities in two inequalities in two using linear inequalities variables and their variables in two variables graphsFormative Determining whether Explaining how to graph an ordered pair is a linear inequalities in two solution to a given variables linear inequality in two variables Differentiating linear inequalities in two Finding the solution set variables from linear of linear inequalities in equations in two two variables variables Solving problems involving linear inequalities in two variables 229

Post-Test: Part I Post-Test: Part I Post-Test: Part I Post-Test: Part I Identifying and Graphing linear Solving problems Products and describing linear inequalities in two involving linear performances related inequalities in two variables inequalities in two to or involving linear variables and their variables inequalities in two graphs Finding the solution set variables of linear inequalities in Representing situations two variables using linear inequalities Part IV: in two variables GRASPS Assessment Summative Part II Part II Part IISelf - assessment Identifying linear Solving linear Describing the solution inequalities in two inequalities in two set of linear inequalities variables variables graphically in two variables and algebraically Part III: Solving problems involving linear inequalities in two variables Journal Writing: Expressing understanding of linear inequalities in two variables Expressing understanding of finding solutions of linear inequalities in two variables graphically and algebraically 230

Assessment Matrix (Summative Test)Levels of Assessment What will I assess? How will I assess? How Will I Score? Paper and Pencil Test 1 point for every correct response The learner demonstrates understanding of key concepts of linear inequalities in two variables. Part I items 2, 4 and 8Knowledge Differentiate between mathematical expressions and Part II item 1 15% mathematical equations. Differentiate between mathematical equations and Part IV item 1 inequalities. Part I items 1, 3, 6, 11, and 12Process/Skills Illustrate linear inequalities in two variables. Part II item 3 1 point for every correct response 25% Rubric on Problem SolvingRu Graph linear inequalities in two variables on the Part I items 5, 7, 9, 13, 14, and 16Understanding coordinate plane. Part II items 2, 4 and 5 1 point for every correct response 30% Solve real-life problems involving linear inequalities in Rubric for explanation two variables. Criteria: Clear Coherent Justified Rubric for drawing Criteria: Neat and Clear Accurate Justified Appropriate Relevant Part III Items 1 and 2 Rubric on Problem Solving The learner is able to formulate real-life problems Part I Items 10, 15, 17, 18, 19, and 1 point for every correct response involving linear inequalities in two variables and 20 solve these with utmost accuracy using a variety of Rubric on Budget Proposal for Raising strategies. Part IV Broiler Chickens GRASPS AssessmentProduct Make a simple budget proposal for Criteria: 30% raising broiler chickens. The budget proposal should be clear, Apply your understanding of the key realistic, and make use of linear concepts of linear inequalities in two inequalities in two variables and other variables in preparing the budget mathematical statements. proposal. 231

C. Planning for Teaching-Learning Introduction: This module covers key concepts of linear inequalities in two variables. It focuses on the three lessons namely: Mathematical Expressions and Equations in Two Variables, Equations and Inequalities in Two Variables, and Graphs of Linear Inequalities in Two Variables. In this module, the students will describe mathematical expressions, mathematical equations and inequalities. They will also illustrate and translate mathematical statements into inequalities. The students will also draw the graphs of linear inequalities in two variables using any graphing materials, tools, or computer software such as GeoGebra. It would be more convenient for students to graph the inequalities in two variables and find its solutions if the use of GeoGebra is encouraged. In all lessons, students are given the opportunity to use their prior knowledge and skills in learning linear inequalities in two variables. They are also given varied activities to process the knowledge and skills learned and to deepen and transfer their understanding of the different lessons. As an introduction to the main lesson, ask them the following questions: Have you asked yourself how your parents budget their income for your family’s needs? How engineers determine the needed materials in the construction of new houses, bridges, and other structures? How students like you spend your time studying, accomplishing school requirements, surfing the internet, or doing household chores? Entice the students to find out the answers to these questions and to determine the vast applications of linear inequalities in two variables through this module. 232

Objectives: After the learners have gone through the lessons contained in this module, they are expected to: a. describe and differentiate mathematical expressions, equations, and inequalities. b. illustrate linear inequalities in two variables using practical situations; c. draw and describe the graphs of linear inequalities in two variables; and d. formulate and solve problems involving linear inequalities in two variables. Teacher’s Note and Reminders Don’t Forget! 233

Pre-Assesment III. PRE - ASSESSMENTCheck students’ prior knowledge, skills, and understanding of mathematics Find out how much you already know about this module. Choose the letter thatconcepts related to Linear Inequalities in Two Variables. Assessing these corresponds to your answer. Take note of the items that you were not able to answerwill facilitate teaching and students’ understanding of the lessons in this correctly. Find the right answer as you go through this module.module. 1. Janel bought three apples and two oranges. The total amount she paid was Answer Key at most Php 123. If x represents the number of apples and y the number of oranges, which of the following mathematical statements represents the givenPart I 6. A 11. C 16. C situation?1. B 7. B 12. B 17. C2. D a. 3x + 2y ≥ 123 c. 3x + 2y > 123 8. B 13. C 18. C b. 3x + 2y ≤ 123 d. 3x + 2y < 1233. B 9. C 14. B 19. B 2. How many solutions does a linear inequality in two variables have?4. D a. 0 b. 1 c. 2 d. Infinite5. C 10. D 15. D 20. B 3. Adeth has some Php 10 and Php 5 coins. The total amount of these coins isTeacher’s Note and Reminders at most Php 750. Suppose there are 50 Php 5-coins. Which of the following is true about the number of Php 10-coins? I. The number of Php 10-coins is less than the number of Php 5-coins. II. The number of Php 10-coins is more than the number of Php 5-coins. III. The number of Php 10-coins is equal to the number of Php 5-coins. a. I and II b. I and III c. II and III d. I, II, and III 4. Which of the following ordered pairs is a solution of the inequality 2x + 6y ≤ 10? a. (3, 1) b. (2, 2) c. (1, 2) d. (1, 0) 5. What is the graph of linear inequalities in two variables? Don’t a. Straight line c. Half-plane Forget! b. Parabola d. Half of a parabola 6. The difference between the scores of Connie and Minnie in the test is not more than six points. Suppose Connie’s score is 32 points, what could be the score of Minnie? a. 26 to 38 b. 38 and above c. 26 and below d. Between 26 and 38 234

Teacher’s Note and Reminders 7. What linear inequality is represented by the graph at the right? a. x – y > 1 b. x – y < 1 c. -x + y > 1 d. -x + y < 1 Don’t 8. In the inequality c – 4d ≤ 10, what could be the possible value of d if c = 8?Forget! a. d ≤ - 1 b. d ≥ - 1 c. d ≤ 1 d. d ≥ 1 2 22 2 9. Mary and Rose ought to buy some chocolates and candies. Mary paid Php 198 for six bars of chocolates and 12 pieces of candies. Rose bought the same kinds of chocolates and candies but only paid less than Php 100. Suppose each piece of candy costs Php 4, how many bars of chocolates and pieces of candies could Rose have bought? a. 4 bars of chocolates and 2 pieces of candies b. 3 bars of chocolates and 8 pieces of candies c. 3 bars of chocolates and 6 pieces of candies d. 4 bars of chocolates and 4 pieces of candies 10. Which of the following is a linear inequality in two variables? a. 4a – 3b = 5 c. 3x ≤ 16 b. 7c + 4 < 12 d. 11 + 2t ≥ 3s 11. There are at most 25 large and small tables that are placed inside a function room for at least 100 guests. Suppose only 6 people can be seated around the large table and only four people for the small tables. How many tables are placed inside the function room? a. 10 large tables and 9 small tables b. 8 large tables and 10 small tables c. 10 large tables and 12 small tables d. 6 large tables and 15 small tables 235

Teacher’s Note and Reminders 12. Which of the following shows the plane divider of the graph of y ≥ x + 4? a. c. b. d. Don’t 13. Cristina is using two mobile networks to make phone calls. One network chargesForget! her Php 5.50 for every minute of call to other networks. The other network charges her Php 6 for every minute of call to other networks. In a month, she spends at least Php 300 for these calls. Suppose she wants to model the total costs of her mobile calls to other networks using a mathematical statement. Which of the following mathematical statements could it be? a. 5.50x + 6y = 300 c. 5.50x + 6y ≥ 300 b. 5.50x + 6y > 300 d. 5.50x + 6y ≤ 300 14. Mrs. Roxas gave the cashier Php 500-bill for three adult’s tickets and five children’s tickets that cost more than Php 400. Suppose an adult ticket costs Php 75. Which of the following could be the cost of a children’s ticket? a. Php 60 b. Php 45 c. Php 35 d. Php 30 236

Teacher’s Note and Reminders 15. Mrs. Gregorio would like to minimize their monthly bills on electric and water consumption by observing some energy and water saving measures. Which of the following should she prepare to come up with these energy and water saving measures? I. Budget Plan II. Previous Electric and Water Bills III. Current Electric Power and Water Consumption Rates a. I and II b. I and III c. II and III d. I, II, and III 16. The total amount Cora paid for two kilos of beef and three kilos of fish is less than Php 700. Suppose a kilo of beef costs Php 250. What could be the maximum cost of a kilo of fish to the nearest pesos? a. Php 60 b. Php 65 c. Php 66 d. Php 67 17. Mr. Cruz asked his worker to prepare a rectangular picture frame such that its perimeter is at most 26 in. Which of the following could be the sketch of a frame that his worker may prepare? a. c. b. d. Don’tForget! 237

Teacher’s Note and Reminders 18. The Mathematics Club of Masagana National High School is raising at least Php 12,000 for their future activities. Its members are selling pad papers and pens to their school mates. To determine the income that they generate, the treasurer of the club was asked to prepare an interactive graph which shows the costs of the pad papers and pens sold. Which of the following sketches of the interactive graph the treasurer may present? a. c. b. d. Don’t 19. A restaurant owner would like to make a model which he can use as a guide inForget! writing a linear inequality in two variables. He will use the inequality in determining the number of kilograms of pork and beef that he needs to purchase daily given a certain amount of money (C), the cost (A) of a kilo of pork, the cost (B) of a kilo of beef. Which of the following models should he make and follow? I. Ax + By ≤ C II. Ax + By = C III. Ax + By ≥ C a. I and II b. I and III c. II and III d. I, II, and III 20. Mr. Silang would like to use one side of the concrete fence for the rectangular pig pen that he will be constructing. This is to minimize the construction materials to be used. To help him determine the amount of construction materials needed for the other three sides whose total length is at most 20 m, he drew a sketch of the pig pen. Which of the following could be the sketch of the pig pen that Mr. Silang had drawn? a. c. b. d. 238

LEARNING GOALS AND TARGETS: WWhhaatt ttoo KKnnooww Students are expected to demonstrate understanding of key concepts of Start the module by assessing your knowledge of the different mathematical linear inequalities in two variables, formulate real-life problems involving concepts previously studied and your skills in performing mathematical operations. these concepts, and solve these with utmost accuracy using a variety of This may help you in understanding Linear Inequalities in Two Variables. As you strategies. go through this module, think of the following important question: “How do linear inequalities in two variables help you solve problems in daily life?” To find out theTopic: Linear Inequalities in Two Variables answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have goneWWhhaatt ttoo KKnnooww over earlier. To check your work, refer to the Answer Key provided at the end of this module. Provide the students opportunity to use some mathematical terms in other Activity 1 WHEN DOES LESS BECOME MORE? contexts by doing Activity 1. Let the students realize that in many real-life situations, mathematical terms are used to compare objects, quantities, Directions: Supply each phrase with the most appropriate word. Explain your and even attributes. Also in this activity, students will be able to recall and answer briefly. familiarize themselves with the terms related to linear inequalities in two variables. QU NS 1. Less money, more __________ 2. More profit, less __________Answer Key 3. More smile, less __________ 4. Less make-up, more __________Activity 1 (Some Possible Answers) 5. More peaceful, less __________1. problems 6. action 6. Less talk, more __________2. investment 7. pest 7. More harvest, less __________3. wrinkles 8. rest 8. Less work, more __________4. beautiful, simple 9. flood 9. Less trees, more __________5. crime 10. wants 10. More savings, less __________ Teacher’s Note and Reminders ?E S T I O a. How did you come up with your answer? b. How did you know that the words are appropriate for the given Don’t Forget! phrases? c. When do we use the word “less”? How about “more”? d. When does less really become more? e. How do you differentiate the meaning of “less” and “less than”? How are these terms used in Mathematics? 239

Present a real-life situation where students could place themselves into it f. How do you differentiate the meaning of “more” and “more than”?and formulate mathematical statements. Ask them to perform Activity 2. In How are these terms used in Mathematics?this activity, the students will be able to see how linear inequalities in two g. Give at least two statements using “less”, “less than”, “more” andvariables are illustrated in real life. There are no specific answers to thequestions in the activity. Students’ responses may vary depending on their “more than”.actual experiences. h. What other terms are similar to the terms “less”, “less than”, Teacher’s Note and Reminders “more” or “more than”? Give statements that make use of these terms. i. In what real-life situations are the terms such as “less than” and “more than” used? How did you find the activity? Were you able to give real-life situations that make use of the terms less than and more than? In the next activity, you will see how inequalities are illustrated in real-life. Activity 2 BUDGET…, MATTERS! Directions: Use the situation below to answer the questions that follow. Amelia was given by her mother Php 320 to buy some food ingredients for “chicken adobo”. She made sure that it is good for 5 people. QU ?E S T I ONS 1. Suppose you were Amelia. Complete the following table with the needed data. Ingredients Quantity Cost per unit Estimated or piece Cost chicken soy sauce vinegar Don’t garlicForget! onion black pepper sugar tomato green pepper potato 240

Teacher’s Note and Reminders 2. How did you estimate the cost of each ingredient? 3. Was the money given to you enough to buy all the ingredients? Justify your answer. 4. Suppose you do not know yet the cost per piece or unit of each ingredient. How will you represent this algebraically? 5. Suppose there are two items that you still need to buy. What mathematical statement would represent the total cost of the two items? From the activity done, have you seen how linear inequalities in two variables are illustrated in real life? In the next activity, you will see the differences between mathematical expressions, linear equations, and inequalities. Activity 3 EXPRESS YOURSELF! Directions: Shown below are two sets of mathematical statements. Use these to answer the questions that follow. y = 2x + 1 y > 2x + 1 3x + 4y = 15 10 – 5y = 7x 3x + 4y < 15 10 – 5y ≥ 7x Don’t y = 6x + 12 9y – 8 = 4x y ≤ 6x + 12 9y – 8 < 4xForget! QU ?E S T I ONS 1. How do you describe the mathematical statements in each set? 2. What do you call the left member and the right member of eachLet the students describe some mathematical statements and ask themto differentiate mathematical expressions, equations, and inequalities. Tell mathematical statement?them to perform Activity 3. Let the students distinguish the different symbols 3. How do you differentiate 2x + 1 from y = 2x + 1? How about 9y – 8used and their meaning in the mathematical statements. Furthermore,emphasize to them that the members on either side of a mathematical and 9y – 8 = 4x?statement are merely expressions. To further strengthen their understanding 4. How would you differentiate mathematical expressions fromof mathematical expressions, equations, and inequalities, ask them to giveand describe some examples of these. mathematical equations? 5. Give at least three examples of mathematical expressions and mathematical equations. 6. Compare the two sets of mathematical statements. What statements can you make? 7. Which of the given sets is the set of mathematical equations? How about the set of inequalities? 8. How do you differentiate mathematical equations from inequalities? 9. Give at least three examples of mathematical equations and inequalities. 241

In Activity 4, let the students identify situations illustrating linear inequalities Were you able to differentiate mathematical expressions from mathematicaland let them write the inequality model. Emphasize that there are cases that equations? How about mathematical equations and inequalities? In the next activity,the word “more than” does not really mean that you will use the symbol “>”. you will identify real-life situations involving linear inequalities.Let them realize also the importance of linear inequality in daily life. Activity 4 “WHAT AM I?”Answer KeyActivity 4 Directions: Identify the situations which illustrate inequalities and write the inequality1. Inequality p < d model in the appropriate column.2. Inequality f > m3. Not g = 1 + 2b Real-Life Situations Classification Inequality Model4. Inequality c ≤ 80 (Inequalities or Not)5. Not w = 46. Inequality g ≥ 75 1. The value of one7. Inequality j < g Philippine peso (p) is less8. Not 7m = f than the value of one US9. Inequality f > c dollar (d)10. Not p = 103 000 000 2. According to the NSO, Teacher’s Note and Reminders there are more female (f) Filipinos than male (m) Don’t Filipinos Forget! 3. The number of girls (g) in the band is one more than twice the number of boys (b). 4. The school bus has a maximum seating capacity (c) of 80 persons 5. According to research, an average adult generates about 4 kg of waste daily (w) 6. To get a passing mark in school, a student must have a grade (g) of at least 75 7. The daily school allowance of Jillean (j) is less than the daily school allowance of Gwyneth (g) 242

Provide the students opportunity to recall and describe graphs of linear 8. Seven times theequations in two variables. Ask them to perform Activity 5. Emphasize that number of malethe graph can be a line that rises to the right if the slope is positive and a line teachers (m) is thethat falls to the right if the slope is negative. This activity will lead students in number of femalelearning how to graph linear inequalities in two variables. teachers (f)Answer Key 9. The expenses for food (f) is greaterActivity 5 than the expenses for1. 4. clothing (c) 10. The population (p) of the Philippines is about 103 000 000 QU ?E S T I ONS 1. How do you describe the situations in 3, 5, 8 and 10? How about the situations in 1, 2, 4, 6, 7 and 9?2. 5. 3. 2. How do the situations in 3, 5, 8 and 10 differ from the situations in 1, 2, 4, 6, 7 and 9? 3. What makes linear inequality different from linear equations? 4. How can you use equations and inequalities in solving real-life problems? From the activity done, you have seen real-life situations involving linear inequalities in two variables. In the next activity, you will show the graphs of linear equations in two variables. You need this skill to learn about the graphs of linear inequalities in two variables. Activity 5 GRAPH IT! A RECALL… Directions: Show the graph of each of the following linear equations in a Cartesian coordinate plane. 1. y = x + 4 2. y = 3x – 1 3. 2x + y = 9 4. 10 – y = 4x 5. y = -4x + 9 243

Let the students identify different points on a given line and describe the QU QU?E S T I ONS NS 1. How did you graph the linear equations in two variables?other points on the Cartesian plane not on the line. Ask them to perform 2. How do you describe the graphs of linear equations in twoActivity 6. In this activity, let the students realize that a line drawn on a planedivides it into two half-planes. Furthermore, deepen their understanding of variables?the solutions of linear equations and the significance of the points that are 3. What is the y-intercept of the graph of each equation? How abouton a given line. Lead the students in understanding linear inequalities in twovariables using the points that are not on the line. the slope? 4. How would you draw the graph of linear equations given the Teacher’s Note and Reminders y-intercept and the slope? Were you able to draw and describe the graphs of linear equations in two variables? In the next task, you will identify the different points and their coordinates on the Cartesian plane. These are some of the skills you need to understand linear inequalities in two variables and their graphs. Activity 6 INFINITE POINTS……… Directions: Below is the graph of the linear equation y = x + 3. Use the graph to answer the following questions. Don’t ?E S T I O 1. How would you describe the line in relation to the plane where it lies?Forget! 2. Name five points on the line y = x + 3. What can you say about the 244 coordinates of these points? 3. Name five points not on the line y = x + 3. What can you say about the coordinates of these points? 4. What mathematical statement would describe all the points on the left side of the line y = x + 3? How about all the points on the right side of the line y = x + 3? 5. What conclusion can you make about the coordinates of points on the line and those which are not on the line?

The succeeding activities are all about linear inequalities in two variables. From the activity done, you were able to identify the solutions of linear equationsBefore the students perform these activities, let them read and understand and linear inequalities. But how are linear inequalities in two variables used in solvingsome important notes on linear inequalities in two variables including their real-life problems? You will find these out in the activities in the next section. Beforegraphs. Tell them to study carefully the examples presented. performing these activities, read and understand first important notes on linear inequalities in two variables and the examples presented. Teacher’s Note and Reminders A linear inequality in two variables is an inequality that can be written in one of the following forms: Ax + By < C Ax + By ≤ C Ax + By > C Ax + By ≥ C where A, B, and C are real numbers and A and B are both not equal to zero. Examples: 1. 4x – y > 1 4. 8x – 3y ≥ 14 2. x + 5y ≤ 9 5. 2y > x – 5 3. 3x + 7y < 2 6. y ≤ 6x + 11 Certain situations in real life can be modeled by linear inequalities. Examples: 1. The total amount of 1-peso coins and 5-peso coins in the bag is more than Php 150. The situation can be modeled by the linear inequality x + 5y > 150, where x is the number of 1-peso coins and y is the number of 5-peso coins. 2. Emily bought two blouses and a pair of pants. The total amount she paid for the items is not more than Php 980. The situation can be modeled by the linear inequality 2x + y ≤ 980, where x is the cost of each blouse and y is the cost of a pair of pants. Don’t The graph of a linear inequality inForget! two variables is the set of all points in the rectangular coordinate system whose ordered pairs satisfy the inequality. When a line is graphed in the coordinate plane, it separates the plane into two regions called half- planes. The line that separates the plane is called the plane divider. 245

Teacher’s Note and Reminders To graph an inequality in two variables, the following steps could be followed. 1. Replace the inequality symbol with an equal sign. The resulting equation becomes the plane divider. Examples: a. y > x + 4 y=x+4 b. y < x – 2 y=x–2 c. y ≥ -x + 3 y = -x + 3 d. y ≤ -x – 5 y = -x – 5 2. Graph the resulting equation with a solid line if the original inequality contains ≤ or ≥ symbol. The solid line indicates that all points on the line are part of the solution of the inequality. If the inequality contains < or > symbol, use a dashed or broken line. The dash or broken line indicates that the coordinates of all points on the line are not part of the solution set of the inequality. a. y > x + 4 c. y ≥ -x + 3 b. y < x – 2 d. y ≤ -x – 5 Don’tForget! 246

Teacher’s Note and Reminders 3. Choose three points in one of the half-planes that are not on the line. Substitute the coordinates of these points into the inequality. If the coordinates of these points satisfy the inequality or make the inequality true, shade the half-plane or the region on one side of the plane divider where these points lie. Otherwise, the other side of the plane divider will be shaded. a. y > x + 4 c. y ≥ -x + 3 For example, points (0, 3), (2, 2), and (4, -5) For example, points (-2, 8), (0, 7), and do not satisfy the inequality y > x + 4. (8, -1) satisfy the inequality y ≥ -x + 3. Therefore, the half-plane that does not Therefore, the half-plane containing contain these points will be shaded. these points will be shaded. The shaded portion constitutes the solution The shaded portion constitutes the of the linear inequality. solution of the linear inequality. b. y < x – 2 d. y ≤ -x – 5 Learn more about Linear Inequalities in Two Variables through the WEB. You may open the following links. 1. http://library.thinkquest. org/20991/alg / systems.html 2. h t t p : / / w w w. k g s e p g . com/project-id/6565- inequalities-two- variable Don’t 3. h t t p : / / w w w .Forget! montereyinstitute.org/ courses/Algebra1/ COURSE_TEXT_ RESOURCE/U05_L2_ T1_text_final.html 4. http://www.phschool. com/atschool/ academy123/english/ academy123_content/ For example, points (0, 5), (-3, 7), and (2, 10) For example, points (12, -3), (0, -9), and (3, -11) do not satisfy the inequality y < x – 2. satisfy the inequality y ≤ -x – 5. wl-book-demo/ph- Therefore, the half-plane that does not Therefore, the half-plane containing these contain these points will be shaded. points will be shaded. 237s.html The shaded portion constitutes the solution The shaded portion constitutes the solution of of the linear inequality. the linear inequality. 5. http://www.purplemath. com/modules/ ineqgrph.html 6. http://math.tutorvista. com/algebra/linear- equations-in-two- variables.html 247

WWhhaatt ttoo PPrroocceessss Now that you learned about linear inequalities in two variables and their graphs, you may now try the activities in the next section.Let the students check their understanding of linear inequalities in two WWhhaatt ttoo PPrroocceessssvariables by doing Activities 7, 8, 9, and 10. Test if they really understood thenotes they have read. Your goal in this section is to learn and understand key concepts of linear inequalities in two variables including their graphs and how they are used in real-lifeAnswer Key situations. Use the mathematical ideas and the examples presented in answering the activities provided.Activity 7 6. Not Linear inequality1. Linear inequality 7. Linear inequality Activity 7 THAT’S ME!2. Not linear inequality 8. Not Linear inequality3. Not linear inequality 9. Not Linear inequality Directions: Tell which of the following is a linear inequality in two variables. Explain your4. Linear inequality 10. Linear inequality answer.5. Linear inequality 1. 3x – y ≥ 12 6. -6x = 4 + 2yActivity 81. Solution 6. Not a solution 2. 19 < y 7. x + 3y ≤ 72. Solution 7. Solution 3. y = 2 x 8. x > -83. Not a solution 8. Solution4. Not a solution 9. Not a solution 55. Solution 10. Not a solution 4. x ≤ 2y + 5 9. 9(x – 2) < 15 Teacher’s Note and Reminders 5. 7(x - 3) < 4y 10. 13x + 6 < 10 – 7y Don’t QU ?E S T I ONS a. How did you identify linear inequalities in two variables? How Forget! about those which are not linear inequalities in two variables? b. What makes a mathematical statement a linear inequality in two variables? c. Give at least three examples of linear inequalities in two variables. Describe each. How did you find the activity? Were you able to identify linear inequalities in two variables? In the next activity, you will determine if a given ordered pair is a solution of a linear inequality. 248

Answer Key Activity 8 WHAT’S YOUR POINT?Activity 8 Directions: State whether each given ordered pair is a solution of the inequality. Justify1. Solution 6. Not a solution your answer.2. Solution 7. Solution3. Not a solution 8. Solution 1. 2x – y > 10; (7, 2) 6. -3x + y < -12; (0, -5)4. Not a solution 9. Not a solution 2. x + 3y ≤ 8; (4, -1) 5. Solution 10. Not a solution 3. y < 4x – 5; (0, 0) 7. 9 + x ≥ y; (-6, 3) 4. 7x – 2y ≥ 6; (-3, -8) Teacher’s Note and Reminders 5. 16 – y > x; (-1, 9) 8. 2y – 2x ≤ 14; (-3, -3) Don’t 9. 1 x + y > 5; (4, 1 ) Forget! 2 2 10. 9x + 2 y < 2; ( 1 ,1) 3 5 QU ?E S T I ONS a. How did you determine if the given ordered pair is a solution of the inequality? b. What did you do to justify your answer?Answer Key From the activity done, were you able to determine if the given ordered pair is a solution of the linear inequality? In the next activity, you will determine if the givenActivity 9 3. a. Yes 5. a. No coordinates of points on the graph satisfy an inequality.1. a. No b. No b. Yes b. Yes c. Yes c. No Activity 9 COME AND TEST ME! c. No d. Yes d. No d. Yes e. Yes e. No Directions: Tell which of the given coordinates of points on the graph satisfy the inequality. e. Yes Justify your answer.2. a. No 4. a. Yes b. Yes b. No 1. y < 2x + 2 c. Yes c. No a. (0, 2) d. Yes d. Yes b. (5, 1) e. Yes e. No c. (-4, 6) d. (8, -9) e. (-3, -12) 249